The Internally 4-Connected Binary Matroids with No \(M(K_{3,3})\)-Minor

Dillon Mayhew, Victoria University of Wellington, New Zealand, Gordon Royle, University of Western Australia, Crawley, Western Australia, and Geoff Whittle, Victoria University of Wellington, New Zealand

The authors give a characterization of the internally \(4\)-connected binary matroids that have no minor isomorphic to \(M(K_{3,3})\). Any such matroid is either cographic, or is isomorphic to a particular single-element extension of the bond matroid of a cubic or quartic Möbius ladder, or is isomorphic to one of eighteen sporadic matroids.